%I #42 Sep 01 2023 11:25:59
%S 0,1,4,14,48,165,572,2002,7072,25194,90440,326876,1188640,4345965,
%T 15967980,58929450,218349120,811985790,3029594040,11338026180,
%U 42550029600,160094486370,603784920024,2282138106804,8643460269248
%N An inverse Chebyshev transform of x^3.
%C The sequence is 0,0,0,1,0,4,0,14,0,...with zeros restored. Second binomial transform of (-1)^n*A003518(n). Second binomial transform of expansion of x^3*c(-x)^8, where c(x) is g.f. of A000108. The g.f. is transformed to x^3 under the Chebyshev transformation A(x) -> (1/(1+x^2))*A(x/(1+x^2)). For a sequence b(n), this corresponds to taking Sum_{k=0..floor(n/2)} C(n-k,k) * (-1)^k * b(n-2k), or Sum_{k=0..n} C((n+k)/2,k) * b(k) * (-1)^((n-k)/2) * (1+(-1)^(n-k))/2.
%C Let X_n be the set of all noncrossing set partitions of an n-element set which either do not contain {n-1,n} as a block, or which do not contain the block {n} whenever 1 and n-1 are in the same block. For n>0, (-1)^n*a(n) gives the value of the Möbius function of X_{n+2} ordered by dual refinement between the discrete and the full partition. For example, X_3 is a chain consisting of 3 elements and its Möbius function between least and greatest element therefore takes the value a(1)=0. - _Henri Mühle_, Jan 10 2017
%H G. C. Greubel, <a href="/A099376/b099376.txt">Table of n, a(n) for n = 0..1000</a>
%H H. Mühle, <a href="https://arxiv.org/abs/1701.02109">Two Posets of Noncrossing Partitions Coming From Undesired Parking Spaces</a>, arXiv:1701.02109 [math.CO], 2017.
%F G.f.: (1-2*x)^4*( sqrt((1+2*x)/(1-2*x)) - 1)^8/(256*x^5).
%F a(n) = Sum_{k=0..n} (k+1)*C(n, (n-k)/2)*(-1)^k*( C(3, k) -3*C(2, k) +3*C(1, k) -C(0, k) )*(1+(-1)^(n-k))/(n+k+2).
%F a(n) = A002057(n-1). - _Michael Somos_, Jul 31 2005
%F Given an ellipse with eccentricity e and major and minor axis a and b respectively, then ((a-b)/ (a+b))^2 = 1*(e/2)^4 +4*(e/2)^6 +14*(e/2)^8 +48*(e/2)^10 + ... - _Michael Somos_, Apr 11 2007
%F E.g.f.: exp(2x)*(Bessel_I(1,2x) - Bessel_I(3,2x)). - _Paul Barry_, Jun 04 2007
%F D-finite with recurrence (n+3)*(n-1)*a(n) -2*n*(2*n+1)*a(n-1)=0. - _R. J. Mathar_, Sep 26 2012
%F a(n) = A000108(n+2) - 2*A000108(n+1) for n>0. - _Henri Mühle_, Jan 10 2017, corrected Sep 25 2021
%F G.f.: ( (1-2*x)*c(x) - (1-x) )/x^2, where c(x) is the gf of A000108. - _G. C. Greubel_, May 05 2021
%F From _Peter Bala_, Aug 30 2023: (Start)
%F a(n) = 2*n/((n+2)*(n+3)) * binomial(2*n+2, n+1).
%F a(n) = 2*Sum_{k = 0..n-1} 1/(n+1)*binomial(n+1, k)*binomial(n+1, k+2). (End)
%t Table[CatalanNumber[n+2] -2CatalanNumber[n+1], {n, 0, 30}] (* or *)
%t Table[4 Binomial[2#+3, #]/(#+4) &[n-1], {n, 0, 30}] (* _Michael De Vlieger_, Jan 10 2017, latter after _Harvey P. Dale_ at A002057 *)
%o (PARI) {a(n)= if(n<1, 0, n++; 2* binomial(2*n, n-2)/n)} /* _Michael Somos_, Apr 11 2007 */
%o (Magma) [Catalan(n+2) -2*Catalan(n+1): n in [0..30]]; // _G. C. Greubel_, May 05 2021
%o (Sage) [catalan_number(n+2) -2*catalan_number(n+1) for n in (0..30)] # _G. C. Greubel_, May 05 2021
%Y Cf. A000108, A000245, A003518, A071718.
%Y Partial sums of A026016.
%K easy,nonn
%O 0,3
%A _Paul Barry_, Oct 13 2004